Chapter 12: Network models PowerPoint Slides Prepared By: Alan Olinsky Bryant University Management Science: The Art of Modeling with Spreadsheets, 2e S.G. Powell K.R. Baker © John Wiley and Sons, Inc.

Introduction  The network model describes patterns of flow in a connected system, where the flow might involve material, people, or funds.  The system elements may be locations, such as cities, warehouses, or assembly lines; or they may be points in time rather than points in space.  When we construct diagrams to represent such systems, the elements are represented by nodes, or circles, in the diagram. The paths of flow are represented by arcs, or arrows.

The Transportation Model  A very common supply chain involves the shipment of goods from suppliers at one set of locations to customers at another set of locations.  The classic transportation model is characterized by a set of supply sources (each with known capacities), a set of demand locations (each with known requirements) and the unit costs of transportation between supply-demand pairs.

Transportation Problem: Model Formulation  The transportation model has two kinds of constraints: capacity constraints and demand constraints  If total capacity equals total demand, both capacity and demand constraints are “=“.  If capacity exceeds demand, the capacity constraints are “ =“.  If demand exceeds capacity, the capacity constraints are “>=” and the demand constraints are “<=“.

Transportation Problem: Spreadsheet Model  It is helpful to depart from the standard linear programming layout of the previous chapter and adopt a special format for this type of model.  In particular, we can construct a model in rows and columns to mirror the table of parameters that describes the problem.  In the Parameters module of the worksheet, all of the unit costs are displayed in an array.  In the Decisions module, the decision variables appear in an array of the same size.  At the right of each decision row is the “Sent” quantity, which is simply the sum of the flows along the row.  These figures align with the capacities given in the Parameters module.  Below each decision column is the “Received” quantity, which is the sum down the column.

Transportation Problem: Example

Excel Mini-Lesson: The SUMPRODUCT Function  The SUMPRODUCT function in Excel takes the pairwise products of two sets of numbers and sums the products. The form of the function is the following: SUMPRODUCT(Array1,Array2) Array1 references the first set of numbers. Array2 references the second set of numbers.

Optimization: Using Solver

Replacing Ranges with Range Names

Sensitivity Analysis  The concepts of sensitivity analysis that were introduced in the previous chapter apply here as well.  The Solver Sensitivity tool may be used with network linear programming models with no new considerations.  We can interpret patterns in the optimal solution in terms of economic priorities.

Sensitivity Analysis on Transportation Problems  In the transportation model, we have supply and demand constraints, and the solution to the model provides shadow prices on each.  The shadow price on a demand constraint tells us how much it costs to ship the marginal unit to the corresponding location, and sometimes, this figure is not obvious without some careful thought.

Process as Applied to Solution of Transportation Problem  Identify a high priority demand—one that is covered by a unique source—and allocate the entire demand to this route. Remove this demand from consideration.  Identify a high priority capacity—one that supplies a single destination—and allocate the entire supply to this route. Remove this supply from consideration.  Repeat the previous two steps using remaining demands and remaining supplies each time, until all shipments are accounted for.

Assignment Model  An important special case of the transportation problem occurs when all capacities and all requirements are equal to one.  In addition, total supply equals total demand.  The classic assignment model is characterized by a set of people, a set of tasks, and a score for each possible assignment of a person to a task.  The problem is to find the best assignment of people to tasks.

Assignment Problem: Spreadsheet Model

Assignment Problem: Sensitivity Analysis  It is rare that we would want to perform sensitivity analysis with respect to either the supply parameters or the demand parameters in an assignment model.  However, we may well be interested in sensitivity analysis with respect to the cost parameters.

The Transshipment Model  The transshipment problem is a more complex version of the transportation problem, characterized by two stages of flow instead of just one.

The Transshipment Model: Graphical Display

Transshipment Problem: Spreadsheet Model

Standard Form for Network Models  It is possible to formulate any of these problems as linear programs built exclusively on balance equations.  Although this approach may not seem as intuitive, it does link the flow diagram and the spreadsheet model more closely, and it allows us to see a more general structure that encompasses other network models as well.

Classic Network Model  In a classical network model, each node in the network corresponds to a material-balance equation—the requirement that total outflow must equal total inflow.  Once we have a network diagram for a problem, we can translate it into a linear program by following these simple steps: Define a variable for each arc. Include supplies as input flows and demands as output flows. Construct the balance equation for each node.

Balance Equations  In this version of the model, every constraint is a balance equation. Thus, the constraints take the following form. (Flow Out) – (Flow In) = 0  We write a balance equation with a positive right-hand side when there is flow into the network and with a negative right-hand side when there is flow out of the network.

Worksheet for the Revised Model

Standard Form Links to Diagram  The standard form has features that link it closely to the diagram: Each variable corresponds to an arc in the diagram (and vice versa.) Each constraint corresponds to a node in the diagram (and vice versa.) The right-hand side constants of the model show up in the network as supplies and demands, with the convention that supply values are positive and demand values negative. The coefficients of a given variable in the constraint equations include only two nonzero values, +1 and –1, which occur in constraints corresponding to the tail and head of the corresponding arc, respectively.

Advantages of Standard Form  The virtue of using standard form is the fact that we can draw on the network diagram as a debugging aid.  This may not seem like a large enough benefit to warrant using the standard form when the classical layout is so intuitive.  However, the concept becomes helpful in models that are more complicated than the transportation model.

Network Models with Yields  If the flows of interest are subject to positive or negative yields, we can incorporate that feature as well.  Waste in a manufacturing process is an example of a negative yield; interest on a bank balance is an example of a positive yield.

Yields as Reductions in Flow  One common type of yield phenomenon involves technologies that produce waste.  For example: When wood is cut, shaped, and sanded in a manufacturing process, the amount of useable wood that exits the process is less than the amount that entered. When metal enters a process that involves grinding, drilling, and polishing, the same is true of the amount of metal at the end of the process compared to the amount at the start.  This type of reduction in the amount of a flow is called process yield.

Yields as Reductions in Flow  There is an intimate relation between the network diagram and the spreadsheet model.  Each arc in the network corresponds to a variable, each node in the network corresponds to a constraint, and all constraints are written as balance equations.  The variables are measured in tons of supply; these are the quantities that are started into the various processes.  To compute the quantities in tons of output, we have to apply yield factors.  The objective function computes the revenue from the schedule of outputs, as supply costs are fixed for the purposes of this decision.

Example

Yields as Expansions in Flow  In production processes, yields are typically less than one: outputs are smaller than inputs because some waste is generated along the way.  However, in other kinds of processes, yields can be greater than one.  An example of this feature arises in funds- flow models.

Investment and Funds-Flow Problems  Investment and funds-flow problems sort lend themselves to network modeling.  In this type of problem, nodes represent points in time at which funds flows occur.  We can imagine tracking a bank account, with funds flowing in and out, depending on our decisions.

Diagram of an Investment Problem

Spreadsheet for an Investment Model

Solver Tip: Rescaling the Model  The way that Solver carries out its arithmetic sometimes makes rescaling desirable.  As a guideline, the parameters in the objective function and the constraints should not differ from each other, or from the values of the decision variables, by more than a factor of 10,000.  A model that tracks cash flows in the millions while also computing percentage returns as decimal fractions violates this rule.  Sometimes, rescaling problems are difficult to avoid when we’re trying to keep the model easy to understand.  In these cases, we can ask Solver to perform some internal rescaling of the model if we check the option box for Use Automatic Scaling.  As helpful as the Automatic Scaling option is, however, it is always preferable for the model builder to do the rescaling.

*7. Network Models for Process Technologies  Some nodes represent production operations that actually transform the substance between inflow and outflow.  This type of network is particularly suitable for the analysis of production plans in process industries, such as paper, steel, or chemicals.

Refining Example  For each node in the diagram, we write a balance equation.  Conceptually, however, there is a twist here.  Because the nodes represent production processes, the input material may differ from the output material, and there may be multiple input materials and multiple output materials.

*Formulation of Refining Example

*Spreadsheet Model of Refining Problem

Summary  Network models represent a distinct class of linear programs.  They have special advantages because network diagrams can be used in the modeling process.  Transportation, assignment, and transshipment models exhibit a characteristic From/To structure that lends itself readily to spreadsheet display.  The balance equations in transshipment nodes are the key constraint format for network models, extending to models in which yield factors apply and even to process models where the inflow and outflow may not be of the same material.  The constraints in some network linear programs consist exclusively of balance equations, whereas constraints in more complicated models may include allocation, covering, and blending constraints appended to a central network representation.

Summary (Continued)  The concepts of sensitivity analysis that were introduced in the previous chapter apply as well to network models.  In particular, when it comes to interpreting optimal solutions, the network diagram is a convenient device for constructing patterns.  The diagram often provides visual hints that lead to a systematic description of the economic priorities in the solution of a network linear program.

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